Noise Wave Correlation Matrix in CAE

Due to the similar concept of S parameters and noise correlation coefficients, the CAE noise analysis can be performed quite alike the S parameter analysis (section 1.3.1). As each step uses the S parameters to calculate the noise correlation matrix, the noise analysis is best done step by step in parallel with the S parameter analysis. Performing each step is as follows: We have the noise wave correlation matrices ( $ (\underline{C})$, $ (\underline{D})$ ) and the S parameter matrices ( $ (\underline{S})$, $ (\underline{T})$ ) of two arbitrary circuits and want to know the correlation matrix of the special circuit resulting from connecting two circuits at one port.

Figure 2.2: connecting two noisy circuits, scheme (left) and signal flow graph (right)

An example is shown in fig. 2.2. What we have to do is to transform the inner noise waves $ \underline{b}_{noise,k}$ and $ \underline{b}_{noise,l}$ to the open ports. Let us look upon the example. According to the signal flow graph the resulting noise wave $ \underline{b}_{noise,i}'$ writes as follows:

$\displaystyle \underline{b}_{noise,i}' = \underline{b}_{noise,i} + \underline{b...
...,l}\cdot \frac{\underline{S}_{ik}}{1-\underline{S}_{kk}\cdot\underline{T}_{ll}}$ (2.25)

The noise wave $ \underline{b}_{noise,j}$ does not contribute to $ \underline{b}_{noise,i}'$, because no path leads to port $ i$. Calculating $ \underline{b}_{noise,j}'$ is quite alike:

$\displaystyle \underline{b}_{noise,j}' = \underline{b}_{noise,j} + \underline{b...
...,k}\cdot \frac{\underline{T}_{jl}}{1-\underline{S}_{kk}\cdot\underline{T}_{ll}}$ (2.26)

Now we can derive the first element of the new noise correlation matrix by multiplying eq. (2.25) with eq. (2.26).

\begin{displaymath}\begin{split}\underline{c}_{ij}' = & \quad \overline{\underli...
...-\underline{S}_{kk}\cdot\underline{T}_{ll} \vert^2} \end{split}\end{displaymath} (2.27)

The noise waves of different circuits are uncorrelated and therefore their time average product equals zero (e.g. $ \overline{\underline{b}_{noise,i}\cdot\underline{b}_{noise,j}^*} =
0$). Thus, the final result is:

\begin{displaymath}\begin{split}\underline{c}_{ij}' = (\underline{c}_{ji}')^* = ...
..._{ik}}{1-\underline{S}_{kk}\cdot\underline{T}_{ll}} \end{split}\end{displaymath} (2.28)

All other cases of connecting circuits can be calculated the same way using the signal flow graph. The results are listed below.

If index $ i$ and $ j$ are within the same circuit, it results in fig. 2.3. The following formula holds:

\begin{displaymath}\begin{split}\underline{c}_{ij}' = (\underline{c}_{ji}')^* = ...
..._{ik}}{1-\underline{S}_{kk}\cdot\underline{T}_{ll}} \end{split}\end{displaymath} (2.29)

This equation is also valid, if $ i$ equals $ j$.

Figure 2.3: connecting two noisy circuits

If the connected ports $ k$ and $ l$ are from the same circuit, the following equations must be applied (see also fig. 2.4) to obtain the new correlation matrix coefficients.

$\displaystyle M = (1-\underline{S}_{kl})\cdot(1-\underline{S}_{lk}) - \underline{S}_{kk}\cdot\underline{S}_{ll}$ (2.30)

$\displaystyle K_1 = \frac{\underline{S}_{il}\cdot(1-\underline{S}_{lk}) + \underline{S}_{ll}\cdot\underline{S}_{ik}} {M}$ (2.31)

$\displaystyle K_2 = \frac{\underline{S}_{ik}\cdot(1-\underline{S}_{kl}) + \underline{S}_{kk}\cdot\underline{S}_{il}} {M}$ (2.32)

$\displaystyle K_3 = \frac{\underline{S}_{jl}\cdot(1-\underline{S}_{lk}) + \underline{S}_{ll}\cdot\underline{S}_{jk}} {M}$ (2.33)

$\displaystyle K_4 = \frac{\underline{S}_{jk}\cdot(1-\underline{S}_{kl}) + \underline{S}_{kk}\cdot\underline{S}_{jl}} {M}$ (2.34)

\begin{displaymath}\begin{split}\underline{c}_{ij}' = \underline{c}_{ij} &+ \und...
...ine{c}_{kl}\cdot K_1 + \underline{c}_{ll}\cdot K_2) \end{split}\end{displaymath} (2.35)

These equations are also valid if $ i$ equals $ j$.

Figure 2.4: connection within a noisy circuits

The absolute values of the noise correlation coefficients are very small. To achieve a higher numerical precision, it is recommended to normalize the noise matrix with $ k\cdot T_0$. After the simulation they do not have to be denormalized, because the noise parameters can be calculated by using equation (2.4) to (2.11) and omitting all occurrences of $ k\cdot T_0$.

The transformer concept to deal with different port impedances and with differential ports (as described in section 1.3.2) can also be applied to this noise analysis.

This document was generated by Stefan Jahn on 2007-12-30 using latex2html.